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Foundations of Vietoris homology theory with applications to non-compact spaces

Seria
Rozprawy Matematyczne tom/nr w serii: 180 wydano: 1980
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Warianty tytułu
Abstrakty
EN
CONTENTS

Preface...................................................................................................................................... 5

I. Introduction............................................................................................................................ 7

II. Simple chains

 2.1. Simplexes............................................................................................................ 12
 2.2. Chains........................................................................................................................... 13
 2.3. Boundary operator. Cycles and boundaries.......................................................... 15
 2.4. Join operator................................................................................................................ 15
 2.5. ε-simplexes and ε-chains........................................................................................... 16

III. Sequential chains

 3.1. Sequences and subsequences...................................................................... 17
 3.2. Sequential chains....................................................................................................... 18
 3.3. Infinite chains. General homology groups............................................................. 18
 3.4. Infinite chains in subspaces..................................................................................... 19
 3.5. True cycles. Vietoris homology groups................................................................... 21
 3.6. Subsequences of infinite chains............................................................................. 22
 3.7. A condition for homology of infinite cycles.............................................................. 24

IV. Functions, mappings, and null translations

4.1. Homomorphisms of simple chains induced by functions...................................... 25
 4.2. Homomorphisms of sequential chains induced by functions........................... 25
 4.3. Homomorphisms of ε-chains induced by functions............................................ 26
 4.4. Homomorphisms of infinite chains induced by maps........................................ 27
 4.5. Topological invariance of the central and Vietoris homology groups............... 28
 4.6. Non-equivalence of the general and Vietoris homology groups....................... 30
 4.7. The homotopy theorem.............................................................................................. 31
 4.8. Null translations.......................................................................................................... 34

V. The Phragmen-Brouwer theorem

 5.1. Introduction.......................................................................................................... 37
 5.2. The Phragmen Brouwer theorem for non-compact spaces............................... 39

VI. The Alexandroff dimension theorem

 6.1. Introduction.......................................................................................................... 40
 6.2. Compactly dimensioned spaces............................................................................. 41
 6.3. The generalized Alexandroff theorem..................................................................... 43

Bibliography.............................................................................................................................. 46
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 180
Liczba stron
47
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CLXXX
Daty
wydano
1980
Twórcy
Bibliografia
  • [1] P. S. Alexandroff, Untersuchungen über Gestalt und Luge abgeschlossener Mengen beliebiger Dimension, Ann. of Math. (2) 30 (1928), pp. 101-187.
  • [2] P. S. Alexandroff, Dimensionstheorie. Ein Beitrag zur Geometrie der abgeschlossenen Mengen, Math. Ann, 106 (1932), pp. 161-238.
  • [3] E. G. Begle, The Vietoris mapping theorem for bicompact spaces, Ann. of Math. 51 (1950), pp. 534-543.
  • [4] R. H. Bing and K. Borsuk, Some remarks concerning topologically homogeneous spaces, ibidem 81 (1965), pp. 100-111.
  • [5] K. Borsuk, Theory of retracts, Monografie Matematyczne 44, PWN - Polish Scientific Publishers, Warsaw 1967.
  • [6] K. Borsuk, Topology of compacta, Lectures given at Rutgers University, New Brunswick, N. J., 1968 (Lecture notes, to appear).
  • [7] K. Borsuk, and A. Kosiński, Families of acyclic compacta in euclidean n-space, Bull. Acad. Polon. Sci. (3) 3 (1955), pp. 293-296.
  • [8] E. Čech, Théorie générale de l'homologie dans un espace quelconque, Fund. Math. 19 (1932), pp. 149-183.
  • [9] J. Dugundji, Topology, Allyn and Bacon, Boston 1966.
  • [10] S. Eilenberg, Sur quelques propriétés des transformations localement homéomorphes, Fund. Math. 24 (1935), pp. 35-42.
  • [11] S. Eilenberg and N. Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, New Jersey 1952.
  • [12] E. E. Floyd, The extension of homeomorphisms, Duke Math. J. 16 (1949), pp. 225-235.
  • [13] J. G. Hocking and G. S. Young, Topology, Addison-Wesley, Reading, Mass., 1961.
  • [14] W. Hurewicz and H. Wallman, Dimension theory, Princeton University Press, Princeton, N, J., 1948.
  • [15] J. L. Kelley, General topology, Van Nostrand, Princeton, N. J., 1955.
  • [16] B. Knaster and C. Kuratowski, Sur les ensembles connexes, Fund, Math. 2 (1921), pp. 206-255.
  • [17] A. Kosiński, On manifolds and r-spaces, ibidem 42 (1955), pp. 111-124.
  • [18] S. Lefschetz, Topology, Amer. Math. Soc. Colloq. Publ. 12, New York 1930.
  • [19] S. Lefschetz, Algebraic topology, ibid. 27, New York 1942.
  • [20] S. Lefschetz, Topics in topology, Ann. of Math. Studies 10, Princeton University Press, Princeton, N. J., 1942.
  • [21] Jun-iti Nagata, Modern dimension theory, North-Holland Publ. Co., Amsterdam 1965.
  • [22] M. H. A. Newman, Local connection in locally compact spaces, Proc. Amer. Math. Soc. 1 (1950), pp. 44-53.
  • [23] D. Rolfsen, Strongly convex metrices in cells. Bull, Amer. Math. Soc. 74 (1968), pp. 171-175.
  • [24] D. Rolfsen, Characterizing the 3-cell by its metric, Fund. Math. 66 (1969), pp. 1-9.
  • [25] S. Smale, A Vietoris mapping theorem for homotopy, Proc. Amer. Math. Soc. 8 (1957), pp. 604-610.
  • [26] E. H. Spanier, Cohomology theory for general spaces, Ann. of Math. 49 (1948), pp. 407-427.
  • [27] N. Steenrod, Regular cycles of compact metric spaces, ibidem 41 (1940), pp. 833-851.
  • [28] L. Vietoris, Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann. 97 (1927), pp. 454-472.
  • [29] G. T. Whyburn, Cyclic elements of higher orders, Amer. J. Math. 56 (1934), pp. 133-146.
  • [30] G. T. Whyburn, The mapping of Betti groups under interior transformations, Duke Math. J. 4 (1938), pp. 1-8.
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Identyfikator YADDA
bwmeta1.element.zamlynska-8f369e96-b1eb-4ec9-99e2-8c366367a5a4
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ISBN
83-01-01116-5
ISSN
0012-3862
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DML-PL
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